[Preface: A while back, Michael Raymer, a professor at the University of Oregon, drew my attention to a curious paper by Craig Holt, who tragically passed away in 2014 [1]. Michael wrote:“Dear Jacques … I would be very interested in knowing your opinion of this paper,since Craig was not a professional academic, and had little community inwhich to promote the ideas. He was one of the most brilliant PhD studentsin my graduate classes back in the 1970s, turned down an opportunity tointerview for a position with John Wheeler, worked in industry until age50 when he retired in order to spend the rest of his time in self study.In his paper he takes a Machian view, emphasizing the relational nature ofall physical quantities even in classical physics. I can’t vouch for thetechnical correctness of all of his results, but I am sure they areinspiring.”

The paper makes for an interesting read because Holt, unencumbered by contemporary fashions, freely questions some standard assumptions about the meaning of `mass’ in physics. Probably because it was a work in progress, Craig’s paper is missing some of the niceties of a more polished academic work, like good referencing and a thoroughly researched introduction that places the work in context (the most notable omission is the lack of background material on dimensional analysis, which I will talk about in this post). Despite its rough edges, Craig’s paper led me down quite an interesting rabbit-hole, of which I hope to give you a glimpse. This post covers some background concepts; I’ll mention Craig’s contribution in a follow-up post. ]

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Imagine you have just woken up after a very bad hangover. You retain your basic faculties, such as the ability to reason and speak, but you have forgotten everything about the world in which you live. Not just your name and address, but your whole life history, family and friends, and entire education are lost to the epic blackout. Using pure thought, you are nevertheless able to deduce some facts about the world, such as the fact that you were probably drinking Tequila last night.

The first thing you notice about the world around you is that it can be separated into objects distinct from yourself. These objects all possess properties: they have colour, weight, smell, texture. For instance, the leftover pizza is off-yellow, smells like sardines and sticks to your face (you run to the bathroom).

While bending over the toilet for an extended period of time, you notice that some properties can be easily measured, while others are more intangible. The toilet seems to be less white than the sink, and the sink less white than the curtains. But how much less? You cannot seem to put a number on it. On the other hand, you know from the ticking of the clock on the wall that you have spent 37 seconds thinking about it, which is exactly 14 seconds more than the time you spent thinking about calling a doctor.

You can measure exactly how much you weigh on the bathroom scale. You can also see how disheveled you look in the mirror. Unlike your weight, you have no idea how to quantify the amount of your disheveled-ness. You can say for sure that you are less disheveled than Johnny Depp after sleeping under a bridge, but beyond that, you can’t really put a number on it. Properties like time, weight and blood-alcohol content can be quantified, while other properties like squishiness, smelliness and dishevelled-ness are not easily converted into numbers.

You have rediscovered one of the first basic truths about the world: all that we know comes from our experience, and the objects of our experience can only be compared to other objects of experience. Some of those comparisons can be numerical, allowing us to say how much more or less of something one object has than another. These cases are the beginning of scientific inquiry: if you can put a number on it, then you can do science with it.

Rulers, stopwatches, compasses, bathroom scales — these are used as reference objects for measuring the `muchness’ of certain properties, namely, length, duration, angle, and weight. Looking in your wallet, you discover that you have exactly 5 dollars of cash, a receipt from a taxi for 30 dollars, and you are exactly 24 years old since yesterday night.

You reflect on the meaning of time. A year means the time it takes the Earth to go around the Sun, or approximately 365 and a quarter days. A day is the time it takes for the Earth to spin once on its axis. You remember your school teacher saying that all units of time are defined in terms of seconds, and one second is defined as 9192631770 oscillations of the light emitted by a Caesium atom. Why exactly 9192631770, you wonder? What if we just said 2 oscillations? A quick calculation shows that this would make you about 110 billion years old according to your new measure of time. Or what about switching to dog years, which are 7 per human year? That would make you 168 dog years old. You wouldn’t feel any different — you would just be having a lot more birthday parties. Given the events of last night, that seems like a bad idea.

You are twice as old as your cousin, and that is true in dog years, cat years, or clown years [2]. Similarly, you could measure your height in inches, centimeters, or stacked shot-glasses — but even though you might be 800 rice-crackers tall, you still won’t be able to reach the aspirin in the top shelf of the cupboard. Similarly, counting all your money in cents instead of dollars will make it a bigger number, but won’t actually make you richer. These are all examples of passive transformations of units, where you imagine measuring something using one set of units instead of another. Passive transformations change nothing in reality: they are all in your head. Changing the labels on objects clearly cannot change the physical relationships between them.

Things get interesting when we consider active transformations. If a passive transformation is like saying the length of your coffee table is 100 times larger when measured in cm than when measured in meters, then an active transformation would be if someone actually replaced your coffee table with a table 100 times bigger. Now, obviously you would notice the difference because the table wouldn’t fit in your apartment anymore. But imagine that someone, in addition to replacing the coffee table, also replaced your entire apartment and everything in it with scaled-up models 100 times the size. And imagine that you also grew to into a giant 100 times your original size while you were sleeping. Then when you woke up, as a giant inside a giant apartment with a giant coffee table, would you realise anything had changed? And if you made yourself a giant cup of coffee, would it make your giant hangover go away?

Or if you woke up as a giant bug?

We now come to one of the deepest principles of physics, called Bridgman’s Principle of absolute significance of relative magnitude, named for our old friend Percy Bridgman. The Principle says that only relative quantities can enter into the laws of physics. This means that, whatever experiments I do and whatever measurements I perform, I can only obtain information about the relative sizes of quantities: the length of the coffee table relative to my ruler, or the mass of the table relative to the mass of my body, etc. According to this principle, actively changing the absolute values of some quantity by the same proportion for all objects should not affect the outcomes of any experiments we could perform.

To get a feeling for what the principle means, imagine you are a primitive scientist. You notice that fruit hanging from trees tends to bob up and down in the wind, but the heavier fruits seems to bounce more slowly than the lighter fruits (for those readers who are physics students, I’m talking about a mass on a spring here). You decide to discover the law that relates the frequency of bobbing motion to the mass of the fruit. You fill a sack with some pebbles (carefully chosen to all have the same weight) and hang it from a tree branch. You can measure the mass of the sack by counting the number of pebbles in it, but you still need a way to measure the frequency of the bobbing. Nearby you hear the sound of water dripping from a leaf into a pond. You decide to measure the frequency by how many times the sack bobs up and down in between drips of water. Now you are ready to do your experiment.

You measure the bobbing frequency of the sack for many different masses, and record the results by drawing in the dirt with a stick. After analysing your data, you discover that the frequency f (in oscillations per water drop) is related to the mass m (in pebbles) by a simple formula:

where k stands for a particular number, say 16.8. But what does this number really mean?

Unbeknownst to you, a clever monkey was watching you from the bushes while you did the experiment. After you retire to your cave to sleep, the monkey comes out to play a trick on you. He carefully replaces each one of your pebbles with a heavier pebble of the same size and appearance, and makes sure that all of the heavier pebbles are the same weight as each other. He takes away the original pebbles and hides them. The next day, you repeat the experiment in exactly the same way, but now you discover that the constant k has changed from yesterday’s value of 16.8 to the new value of 11.2. Does this mean that the law of nature that governs the bobbing of things hanging from the tree has changed overnight? Or should you decide that the law is the same, but that the units that you used to measure frequency and mass have changed?

You decide to apply Bridgman’s Principle. The principle says that if (say) all the masses in the experiment were changed by the same proportion, then the laws of physics would not allow us to see any difference, provided we used the same measuring units. Since you do see a difference, Bridgman’s Principle says that it must be the units (and not the law itself) that has changed. `These must be different pebbles’ you say to yourself, and you mark them by scratching an X onto them. You go out looking for some other pebbles and eventually you find a new set of pebbles which give you the right value of 16.8 when you perform the experiment. `These must be the same kind of pebbles that I used in the original experiment’ you say to yourself, and you scratch an O on them so that you won’t lose them again. Ha! You have outsmarted the monkey.

Notice that as long as you use the right value for k — which depends on whether you measure the mass using X or O pebbles — then the abstract equation (1) remains true. In physics language, you are interpreting k as a dimensional constant, having the dimensions of frequency times √mass. This means that if you use different units for measuring frequency or mass, the numerical value of k has to change in order to preserve the law. Notice also that the dimensions of k are chosen so that equation (1) has the same dimensions on each side of the equals sign. This is called a dimensionally homogeneous equation. Bridgman’s Principle can be rephrased as saying that all physical laws must be described by dimensionally homogeneous equations.

Bridgman’s Principle is useful because it allows us to start with a law expressed in particular units, in this case `oscillations per water-drop’ and `O-pebbles’, and then infer that the law holds for any units. Even though the numerical value of k changes when we change units, it remains the same in any fixed choice of units, so it represents a physical constant of nature.

The alternative is to insist that our units are the same as before (the pebbles look identical after all). That means that the change in k implies a change in the law itself, for instance, it implies that the same mass hanging from the tree today will bob up and down more slowly than it did yesterday. In our example, it turns out that Bridgman’s Principle leads us to the correct conclusion: that some tricky monkey must have switched our pebbles. But can the principle ever fail? What if physical laws really do change?

Suppose that after returning to your cave, the tricky monkey decides to have another go at fooling you. He climbs up the tree and whispers into its leaves: `Do you know why that primitive scientist is always hanging things from your branch? She is testing how strong you are! Make your branches as stiff and strong as you can tomorrow, and she will reward you with water from the pond’.

The next day, you perform the experiment a third time — being sure to use your `O-pebbles’ this time — and you discover again that the value of k seems to have changed. It now takes many more pebbles to achieve a given frequency than it did on the first day. Using Bridgman’s Principle, you again decide that something must be wrong with your measuring units. Maybe this time it is the dripping water that is wrong and needs to be adjusted, or maybe you have confidence in the regularity of the water drip and conclude that the `O-pebbles’ have somehow become too light. Perhaps, you conjecture, they were replaced by the tricky monkey again? So you throw them out and go searching for some heavier pebbles. You find some that give you the right value of k=16.8, and conclude that these are the real `O-pebbles’.

The difference is that this time, you were tricked! In fact the pebbles you threw out were the real `O-pebbles’. The change in k came from the background conditions of the experiment, namely the stiffness in the tree branches, which you did not consider as a physical variable. Hence, in a sense, the law that relates bobbing frequency to mass (for this tree) has indeed changed [3].

You thought that the change in the constant k was caused by using the wrong measuring units, but in fact it was due to a change in the physical constant k itself. This is an example of a scenario where a physical constant turns out not to be constant after all. If we simply assume Bridgman’s Principle to be true without carefully checking whether it is justified, then it is harder to discover situations in which the physical constants themselves are changing. So, Bridgman’s Principle can be thought of as the assumption that the values of physical constants (expressed in some fixed units) don’t change over time. If we are sure that the laws of physics are constant, then we can use the Principle to detect changes or inaccuracies in our measuring devices that define the physical units — i.e. we can leverage the laws of physics to improve the accuracy of our measuring devices.

We can’t always trust our measuring units, but the monkey also showed us that we can’t always trust the laws of physics. After all, scientific progress depends on occasionally throwing out old laws and replacing them with more accurate ones. In our example, a new law that includes the tree-branch stiffness as a variable would be the obvious next step.

One of the more artistic aspects of the scientific method is knowing when to trust your measuring devices, and when to trust the laws of physics [4]. Progress is made by `bootstrapping’ from one to the other: first we trust our units and use them to discover a physical law, and then we trust in the physical law and use it to define better units, and so on. It sounds like a circular process, but actually it represents the gradual refinement of knowledge, through increasingly smaller adjustments from different angles. Imagine trying to balance a scale by placing handfuls of sand on each side. At first you just dump about a handful on each side and see which is heavier. Then you add a smaller amount to the lighter side until it becomes heavier. Then you add an even smaller amount to the other side until it becomes heavier, and so on, until the scale is almost perfectly balanced. In a similar way, switching back and forth between physical laws and measurement units actually results in both the laws and measuring instruments becoming more accurate over time.

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[1] It is a shame that Craig’s work remains incomplete, because I think physicists could benefit from a re-examination of the principles of dimensional analysis. Simplified dimensional arguments are sometimes invoked in the literature on quantum gravity without due consideration for their meaning.

[2] Clowns have several birthdays a week, but they aren’t allowed to get drunk at them, which kind of defeats the purpose if you ask me.

[3] If you are uncomfortable with treating the branch stiffness as part of the physical law, imagine instead that the strength of gravity actually becomes weaker overnight.

[4] This is related to a deep result in the philosophy of science called the Duhem-Quine Thesis.
Quoth Duhem: `If the predicted phenomenon is not produced, not only is the questioned proposition put into doubt, but also the whole theoretical scaffolding used by the physicist’.

This is one of those questions that has always bugged me.
Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?

“I know who I WAS when I got up this morning, but I think I must have been changed several times since then.”
— Alice Through the Looking-Glass

Magritte – La Reproduction Interdite

Professor Lee Tsung gripped the edge of the sink and stared into the eyes of her reflection. In the Physics department’s toilets, all was silent, except for the dripping of a single tap, and her terse breathing. Finally, the door swung open and someone else entered, breaking the spell. The intruder was momentarily baffled by the end of a phrase that Lee whispered to herself on her way out. Something about her eyes being switched.

“I’m a mess,” she confessed later to Yang Chen, her friend and colleague in the particle physics department. Both women had been offered jobs at the American Institute of Particle Physics at the same time, and had become close friends and collaborators in their work on extensions to the Standard Model. Lee Tsung’s ambition and penetrating vision had overflowed into her personal life. It infiltrated her mannerisms, giving her nervous ticks, fiery and unstable relationships and a series of dramatic break-downs and comebacks that littered her increasingly illustrious career. Chen was the antidote to Lee’s anxious and hyperactive working style; calm, meditative and thorough, her students’ biggest complaint was that she was boring. But united by a common goal, the two scientists had worked perfectly together to uncover some of the deepest secrets of nature, pioneering the field of research on neutrino oscillations and being heavily involved with experimentalists and theoreticians in the recently successful hunt for the Higgs Boson.

“I have to quit physics. I can’t look at another textbook. I don’t dare.”

Chen was visibly shocked. In all of the ups and downs she had experienced with Tsung, the idea of quitting physics altogether had never arisen.

“What’s going on? What happened?”

Lee fixed her friend with a two-colour gaze. Her eyes, normally hidden in brooding black shadows beneath her eyebrows, now reflected the dim light of the university cafe. One eye was green and the other brown.

“You’ll think I’m crazy.”

“I already do. You’ve got nothing to lose.”

Lee studied her friend a moment longer, then sighed in resignation.

“Well if I don’t tell you, I suppose I’ll go mad anyway. But before I tell you, you have to promise me something. Promise me that you won’t tell me, not even drop a hint, which way the bias went in the Wu experiment.”

“What, you mean the parity violation experiment?”

“Don’t even talk about it! It is too dangerous. You might let something slip. I don’t want to know anything about their result.”

“But you already know the result — you lectured on particle theory back when you were an associate professor.”

“Shush already! Just listen. Last night I was working late at the cyclotron. Nothing to do with the machine itself, I was just hanging out in the lab trying to finish these damn calculations — that blasted theoretical paper on symmetry breaking. Anyway, I could barely keep my eyes open. You know how the craziest ideas always come just when you are dozing off to sleep? Well, I had one of those, and it was, –” Lee broke off to give a short laugh, “– well, it was as crazy as they come. I was thinking, isn’t it weird that the string theorists tell us every particle has a heavier super-partner, so that their calculations balance out. Well, why don’t we do the same thing for parity?”

“You lost me already.”

“You know, what if there is another universe that exists, identical to ours in every way — except that it is flipped. Into its mirror image.”

“Like in Alice through the looking-glass?”

“Exactly. In theory, there is nothing against having matter that is ordinary in every respect, except with left and right handedness reversed. The only question would be, if such a mirror-image universe existed, where is it? So I did some back-of-the-envelope calculations and found out that the coupling between mirror-matter and ordinary matter would be extremely weak. The two worlds could co-exist side by side, and we wouldn’t even know!”

Chen stirred her coffee, looking almost bored, but she didn’t drink it for a full minute. In any case, it didn’t need stirring — there was no sugar in it. Lee could almost see her friend’s mind working. At last she said:

“That’s pretty unlikely. We have some really sensitive instruments downstairs, and some really high energies. Are you telling me even they wouldn’t be able to pick something up?”

Lee’s eyes glittered with excitement.

“Exactly! That’s what I thought. It turns out that, with just a small modification of the collider experiment — the one we’re running right this minute — we could instantiate a reaction between our universe and the mirror universe. We could even exchange a substantial amount of matter between the two worlds!”

Chen’s looked up sharply.

“Lee,” she said, “Please tell me you didn’t already do it.”

Lee shrugged guiltily.

“It only cost them a few hours of data. Nobody will care – they’ll assume the diversion of the beam was due to a mistake by one of the grad students. But it worked Chen! I opened up the mirror world.”

“That’s great! You’ll be famous!”

But Lee stared down at her fingers, the nails chewed raw, and said nothing.

“That’s a Nobel prize right there! Why are you freaking out?”

“Just wait. We’re about to get to the really weird part,” said Lee.

“At first I thought the experiment wasn’t working. I had expected to see a definite interaction region, generating all kinds of particles. But I guess I hadn’t really worked out the details of what to expect. Nothing seemed to be happening to the beam. I was puzzled for a few seconds. But then there was a weird change in the light of the whole building – it seemed to get slightly brighter. And everything doubled — even the humming of the machine seemed to get twice as loud. And the numbers coming up were really bizarre. I started seeing not just the normal data stream, but additional characters superimposed on top. I don’t know that much about the displays we have, but I’m pretty sure they can’t make figures like that.”

“What do you mean?”

“They were backwards. You know, like when you try to read a book in the mirror. When I looked at the clock on the wall, it had six hands, and the two second-hands were ticking in opposite directions. The three wrong hands seemed to flicker in and out of existence, along with the extra light and sound that was coming through. ”

“Yes. Well, at least the lab and most of the surrounding infrastructure. I don’t know if it extended outside. Nobody would have noticed, I think. It was 3am.”

Chen exhaled and leaned back in her chair.

“Wow. Okay, that does sound pretty nuts. In fact, it sounds just like a crazy dream. Have you seen a therapist?”

“It felt pretty real to me. And it gets crazier. I got this overwhelming sense of dread. Fear, like you can’t imagine. I couldn’t figure out where it was coming from, and then I realised that my PC screen had gone dark, and I could see my own reflection in it.”

She swallowed.

“Not just everything in the room, but me too — I was also doubled up. There were two versions of me coexisting at once. One of them was facing me, but the other, sitting in the same place — I could only see the back of her head. I don’t know why I felt so afraid. But I did.”

“But if the two worlds were really superimposed on each other like you said, and the interaction region was so large, than the coupling must have still been very weak. Probably mostly electromagnetic. So it would have ended the party pretty fast, right? And with a very low rate of matter exchange.”

“Yes, that’s what I expected too. But it didn’t happen. The double-images persisted for at least a minute. It seemed like forever, just me sitting there staring at the back of my own head bizarrely superimposed on my face. I kept wondering, if I were in the other world, would I also just be sitting here? But it was the strangest thing … it was quiet enough that I could hear us both breathing. Her and me — or me and me, I suppose. But I could also hear buttons being pressed at the console, very slowly and softly. I recognised the `click’ they made. It was unmistakable. I swear she had an arm up somewhere where I couldn’t see in the reflection, controlling the beam.”

“Wait. What does that mean? What are you saying?”

Lee leaned forward, almost hesitating to speak. When she did, it was in a low voice, uncharacteristic of her.

“Don’t play dumb Chen. I’m saying, maybe that interaction was exactly as strong as I expected. But maybe all the matter exchange was happening at a single location. It’s against protocol, but I know that in principle we can localise the beam almost anywhere within the facility … including the console where I was sitting.”

“You think your mirror-self was focusing the interaction on herself? On you? Swapping your matter with hers between the two universes?”

“Well, I don’t know really – I sort of blacked out. I woke up after 5am, slumped at the desk by the beam controls, drooling like a baby. First thing I did was look at the clock. The second hand was moving clockwise, like always. I checked every document I could find, it was all written left to right. I still have the birthmark on my left hand, not the right. You can verify that.”

Lee held up her hand. Chen said:

“So, everything’s fine then! You’re not in the mirror world. You’re in the real world. I’m the real me, you’re the real you. And you’ll be famous for discovering mirror matter!”

Lee shook her head.

“There’s more. I was looking at myself in the mirror just before I called you today. Staring at my eyes. Of course, as you might guess, the left one was still brown and the right one was still green, as it always was. But a thought occurred to me. These eyes being examined — they themselves are the examiners. The world looks the right way around, alright, but what if I’m seeing it through mirrored eyes? Sure, when you look in the mirror everything looks flipped left and right from your perspective. But what if your perspective also gets flipped?”

“You mean, if a mirror image reads a book, does it look backwards to her?”

“Exactly. You look at your reflection and you see its left and right hands are switched. But if you point to the hand that you think is your right hand, the image points to its left hand. The image perceives it’s left hand as its right hand. It looks at you, and from its perspective, you are the one who has got it wrong. The mirror image can’t tell that it is a mirror image.”

Chen was silent.

“Imagine that, next time you wake up, you have a fifty-fifty chance of waking up as your own mirror image, in the mirror universe. Could you leave any kind of trace or signal for yourself that you could use to tell if it happened or not? You can try to use books, clocks, birthmarks or even tattoo yourself with big signs saying `left hand’ and `right hand’ — but it won’t do any good, because the switch is always compensated by your own switch in perspective. You lose your reference point.”

“Come on. There has to be some way to tell. Gyroscopes? Spinning stars? Light polarisation?”

Lee shook her head.

“Not quite. We know that classical physics is parity invariant. Newton’s equations, even electromagnetism, all invariant under flipping left and right. Think more fundamental. More modern.”

Chen sucked on the end of her spoon, frowning at the ceiling of the cafe. She still hadn’t touched her coffee. Then she gasped:

“Beta decay! Weak interactions violate parity! You could just dig up the paper by Wu and check whether the direction of emission was — mmf!”

Lee lunged over the table, upsetting a glass of water, to block Chen’s mouth forcibly with her hand.

“Sorry!” Chen whispered breathlessly. Lee glared at her.

“I told you! I don’t want to know which way those experiments came out.”

There was a pause as a waiter came to mop up the spill, glancing nervously at the two women before moving on.

“But why not?” said Chen when he was gone.

“It will resolve your question once and for all. In fact, –” she rummaged around in her bag, “I have my lecture notes here that I prepared for the particle physics course. It’s on page 68 … or maybe 69. One of those.”

She carefully placed the paper on the table, her lips pursed to prove to Lee that she wasn’t going to accidentally mention the answer. Lee took the paper and placed her coffee cup solidly on top.

“You have to understand,” said Lee, “what it would mean if I was right. If I find out that the direction of parity violation in this universe is different to what it is in my memory, that means that I live in the wrong world. Could I live with that knowledge?”

Chen shrugged.

“Why not? You said everything is the same between the worlds, right? Who cares if your left hand turned into your right?”

“Yes. That’s the point. Doesn’t it all seem rather convenient? That the mirror world should exactly match our universe, in all details, apart from things like the direction of beta decay. Don’t forget, we are talking about physics here. Simply reversing the handedness of matter doesn’t say anything about what kind of world is constructed out of it. In all probability, I should have coupled through to just empty space — after all, most of the universe is empty space. Why should the orbits of the two Earth’s match up so precisely, and their rotations, right down to the location of my laboratory? Think about it Chen. What kind of powerful force could arrange everything just so that, if one person should find a way to open up a gateway between the worlds, it would be possible to switch her with her mirror copy, and she wouldn’t notice a thing?”

“I don’t know, Lee!” said Chen, now visibly exasperated.

Lee slumped in her chair.

“I don’t know either,” she said at last.

“That’s the thing that bothers me. Maybe there really is a perfectly good reason why there should be another copy of Earth in the mirror universe, and another copy of me. But why would the people on the other side try to hide it? It is like they were waiting for it. Like they didn’t want me to know the difference. So that I wouldn’t try to go back after the switch was made. But if that’s true, then who — or what — took my place?”

Chen held her friend’s hand tight.

“Okay. I believe you. But please listen to me. It might seem like a fifty-fifty chance whether you are in the real world or the mirror world. But that’s not how probability works. Taking into account human psychology, you have to admit, it’s far more likely that you are suffering from delusions. I beg you, read the results of the Wu experiment. Set your mind at rest. The odds aren’t even — they’re a billion to one against. Do it right now, and end this.”

Lee looked at the document in front of her. She sighed. What was worse, knowing or not knowing? She leafed over to page 68.

“Hmm.”

Chen waited eagerly.

“Just preliminaries.”

“Must be the next page!”

Chen nearly tore the page while turning it, and pointed to the relevant paragraph.

“Look. That’s where I describe the set-up.”

Lee was looking curiously at her friend.

“What? Don’t you want to know?”

“I don’t know. Why are you so eager to show me? Is this part of the plan?”

Chen was caught off guard.

“What plan? Come on, Lee. I just want you to stop acting crazy so we can go back to how things were before.”

Lee held Chen’s eyes a moment longer, then finally looked down at the page.

“Okay. From what I remember, the electrons were biased parallel to the direction of current through the coil.”

She skimmed through the paragraph with her fingertip and then froze. For a moment, she seemed unable to lift her head, and when she did, her eyes held a peculiar expression.

“Chen…” she said, and her voice barely came out, “this says the bias was against the current.”

A strange blank stare had come over Chen’s face.

“Who … what are you?” said Lee. Her voice rose and her eyes flickered over the people around her in the cafe:

“Why did you do this? What world is this?”

She stood up and her chair scraped backwards and almost toppled. At the sound, Chen’s peculiar paralysis seemed to break. She held up both her hands.

“Woah woah woah. Calm down.”

“What! Explain to me what the hell is going on!”

“Listen to me,” said Chen urgently, “what charge convention were you using?”

Lee stared, speechless. Chen blustered on:

“I know you like to be unconventional. I remember you often saying that sometimes you define current as being in the direction of positive charge, just to keep your students on their toes. If so, then your memory of the result would be that the electrons were emitted parallel to positive current — which means against the actual flow of electrons. Just like it says in my notes.”

Lee stood still for a full ten seconds, and the entire cafe had gone quiet, waiting for a scene to break out. But after what seemed like an eternity, she fell back into her seat and buried her face in her arms.

“I can’t remember.”

And all the while, as Chen tried to console her, as Lee slowly came to terms with the fact that with the one escaped memory of that lecture she had given, ten years ago, where she may or may not have used an unconventional definition of electric current, another memory kept returning to her. It was a memory, not form ten years ago, but in fact from that very same day, after she had woken up — from what might have been a dream — and staggered down to the bathroom to stare at herself so long in the mirror. The memory of what might have been an involuntary twitch of the eye of her reflection. Nothing to notice at the time, but on reflection, now that she thought about it, it seemed almost, unmistakably: a suppressed wink.

As you might have guessed from posts like this one, I am a huge fan of the technique of using a fictional story to get across an idea or a concept. The following are links to some of my favourite examples of this underrated art form, more or less in order of preference.

2. Nick Bostrom, The Fable of the Dragon-Tyrant. I’ve mentioned Bostrom before, but I was unaware of his storytelling powers until I came across this gem. Here, he weaves a story that cleverly gets the reader on his side, before drawing back the curtain to show us what was really at stake the whole time.

Finally, although it is in a somewhat different vein to the above links, I have to mention the work of writer Greg Egan, which epitomizes the concept of “hard sci-fi”: flights of the imagination conceived not only in the spirit of modern science, but in its very clothing. Excerpts from his novels and complete versions of his short stories can be read online here.

One of my favourite mind-bending topics is probability theory. It turns out that, for some reason, human beings are very bad at grasping how probability works. This is evident in many phenomena: why do we think the roulette wheel is more likely to come up black after a long string of reds? Why do people buy lottery tickets? Why is it so freakin’ hard to convince people to switch doors in the famous Monty Hall Dilemma?

Part of the problem is that we seem to think we understand probability much better than we actually do. This is why card sharks and dice players continue to make a living by swindling people who fall into common traps. Studying probability is one of the most humbling things a person can do. One area that has particular relevance to physics is the concept of anthropic reasoning. We base our decisions on prior knowledge that we possess. But it is not always obvious which prior knowledge is relevant to a given problem. There may be some cases where the mere knowledge that you exist — in this time, as yourself – might conceivably tell you something useful.

The anthropic argument in cosmology and physics is the proposal that some observed facts about the universe can be explained simply by the fact that we exist. For example, we might wonder why the cosmological constant is so small. In 1987, Steven Weinberg argued that if it were any bigger, it would not have been possible for life to evolve in the universe — hence, the mere fact that we exist implies that the value of the constant is below a certain limit. However, one has to be extremely careful about invoking such principles, as we will see.

This blog post is likely to be the first among many, in which I meditate on the subtleties of probability. Today, I’d like to look at an old chestnut that goes by many names, but often appears in the form of the `Adam and Eve’ paradox.

Spranger – Adam and Eve

Adam finds himself to be the first human being. While he is waiting around for Eve to turn up, he is naturally very bored. He fishes around in his pocket for a coin. Just for a laugh, he decides that if the coin comes up heads, he will refuse to procreate with Eve, thereby dooming the rest of the human race to non-existence (Adam has a sick sense of humour). However, if the coin comes up tails, he will conceive with Eve as planned and start the chain of events leading to the rest of humanity.

Now Adam reasons as follows: `Either the future holds a large number of my future progeny, or it holds nobody else besides myself and Eve. If indeed it holds many humans, then it is vastly more likely that I should have been born as one of them, instead of finding myself rather co-incidentally in the body of the first human. On the other hand, if there are only ever going to be two people, then it is quite reasonable that I should find myself to be the first one of them. Therefore, given that I already find myself in the body of the first human being, the coin is overwhelmingly likely to come up heads when I flip it.’ Is Adam’s reasoning correct? What is probability of the coin coming up heads?

As with many problems of a similar ilk, this one creates confusion by leaving out certain crucial details that are needed in order to calculate the probability. Because of the sneaky phrasing of the problem, however, people often don’t notice that anything is missing – they bring along their own assumptions about what these details ought to be, and are then surprised when someone with different assumptions ends up with a different probability, using just as good a logical argument.

Any well-posed problem has an unambiguous answer. For example, suppose I tell you that there is a bag of 35 marbles, 15 of which are red and the rest blue. This information is now sufficient to state the probability that a marble taken from the bag is red. But suppose I told you the same problem, without specifying the total number of marbles in the bag. So you know that 15 are red, but there could be any number of additional blue marbles. In order to figure out the probability of getting a red marble, you first have to guess how many blue marbles there are, and in this case (assuming the bag can be infinitely large) a guess of 20 is as good as a guess of 20000, but the probability of drawing a red marble is quite different in each case. Basically, two different rational people might come up with completely different answers to the question because they made different guesses, but neither would be any more or less correct than the other person: without additional information, the answer is ambiguous.

In the case of Adam’s coin, the answer depends on things like: how do souls get assigned to bodies? Do you start with one soul for every human who will ever live and then distribute them randomly? If so, then doesn’t this imply that certain facts about the future are pre-determined, such as Adam’s decision whether or not to procreate? We will now see how it is possible to choose two different contexts such that in one case, Adam is correct, and in the other case he is wrong. But just to avoid questions of theological preference, we will rephrase the problem in terms of a more real-world scenario: actors auditioning for a play.

Imagine a large number of actors auditioning for the parts in the Play of Life. Their roles have not yet been assigned. The problem is that the director has not yet decided which version of the play he wishes to run. In one version, he only needs two actors, while in the other version there is a role for every applicant.

In the first version of the play, the lead actor flips a coin and it comes up heads (the coin is a specially designed stage-prop that is weighted to always come up heads). The lead actress then joins the lead actor onstage, and no more characters are required. In the second version of the play, the coin is rigged to come up tails, and immediately afterwards a whole ensemble of characters comes onto the scene, one for every available actor.

The director wishes to make his decision without potentially angering the vast number of actors who might not get a part. Therefore he decides to use an unconventional (and probably illegal) method of auditioning. First, he puts all of the prospective actors to sleep; then he decides by whatever means he pleases which version of the play to run. If it is the first version, he randomly assigns the roles of the two lead characters and has them dressed up in the appropriate costumes. As for all the other actors who didn’t get a part, he has them loaded into taxis and sent home with an apologetic letter. If he decides on the second version of the play, then he assigns all of the roles randomly and has the actors dressed up in the costumes of their characters, ready to go onstage when they wake up.

Now imagine that you are one of the actors, and you are fully aware of the director’s plan, but you do not know which version of the play he is going to run. After being put to sleep, you wake up some time later dressed in the clothing of the lead role, Adam. You stumble on stage for the opening act, involving you flipping a coin. Of course, you know coin is rigged to either land heads or tails depending on which version of the play the director has chosen to run. Now you can ask yourself what the probability is that the coin will land heads, given that you have been assigned the role of Adam. In this case, hopefully you can convince yourself with a bit of thought that your being chosen as Adam does not give you any information about the director’s choice. So guessing that the coin will come up heads is equally justified as guessing that it will come up tails.

Let us now imagine a slight variation in the process. Suppose that, just before putting everyone to sleep, the director takes you aside and confides in you that he thinks you would make an excellent Adam. He likes you so much, in fact, that he has specially pre-assigned you the role of Adam in the case that he runs the two-person version of the play. However, he feels that in the many-character version of the play it would be too unfair not to give one of the other actors a chance at the lead, so in that case he intends to cast the role randomly as usual.

Given this extra information, you should now be much less surprised at waking up to find yourself in Adam’s costume. Indeed, your lack of surprise is due to the fact that your waking up in this role is a strong indication that the director went with his first choice – to run the two-person version of the play. You can therefore predict with confidence that your coin is rigged to land heads, and that the other actors are most probably safely on their way home with apologetic notes in their jacket pockets.

What is the moral of this story? Be suspicious of any hypothetical scenario whose answer depends on mysterious unstated assumptions about how souls are assigned to bodies, whether the universe is deterministic, etc. Different choices of the process by which you find yourself in one situation or another will affect the extent to which your own existence informs your assignation of probabilities. Specifying these details means asking the question: what process determines the state of existence in which I find myself? If you want to reason about counterfactual scenarios in which you might have been someone else, or not existed at all, then you must first specify a clear model of how such states of existence come about. Without that information, you cannot reliably invoke your own existence as an aid to calculating probabilities.

(Update: My colleagues pointed out that Wittgenstein was one of the greatest philosophers of the 20th century and I should not make fun of him, and anyway he was only very loosely associated with the Vienna circle. All well and true — but he was at least partly responsible for the idea that got the Vienna Circle onto Verificationism, and all of you pedants can go look at the references if you don’t believe me.)

Some physicists give philosophy a bad rap. I like to remind them that all the great figures in physics had a keen interest in philosophy, and were strongly influenced by the work of philosophers. Einstein made contributions to philosophy as well as physics, as did Ernst Mach, whose philosophical work had a strong influence on Einstein in formulating his General Theory of Relativity. In his own attitude to philosophy, Einstein was a self-described “epistemological opportunist” [1]. (Epistemology is, broadly speaking, the philosophy of knowledge and how it is acquired.) But philosophy sometimes gets in the way of progress, as explained in the following story.

A physicist was skipping along one day when he came upon a philosopher, standing rigid in the forest. “Why standeth you thus?” he inquired.

“I am troubled by a paradox!” said the philosopher. “How is it that things can move from place to place?”

“What do you mean? I moved here by skipping, didn’t I?”

“Yes, sure. But I cannot logically explain why the world allows it to be so. You see, a philosopher named Zeno argued that in order to traverse any finite distance, one would have to first traverse an infinite number of partitions of that distance. But how can one make sense of completing an infinite number of tasks in a finite amount of time?”

“Well dang,” said the physicist “that’s an interesting question. But wait! Could it be that space and time are actually divided up into a finite number of tiny chunks that cannot be sub-divided further? What an idea!”

“Ah! Perhaps,” says the philosopher, “but what if the world is indeed a continuum? Then we are truly stuck.”

At that moment, a mathematician who had been dozing in a tree fell out and landed with a great commotion.

“Terribly sorry! Couldn’t help but overhear,” he said. “In fact I do believe it is conceptually possible for an infinite number of things to add up to a finite quantity. Why, this gives me a great idea for calculating the area under curves. Thank you so much, I’d better get to it!”

“Yes, yes we must dash at once! There’s work to do!” agreed the physicist.

“But wait!” cried the philosopher, “what if time is merely an illusion? And what is the connection of abstract mathematics to the physical world? We have to work that out first!”

But the other two had already disappeared, leaving the philosopher in his forest to ponder his way down deeper and ever more complex rabbit-holes of thought.

***

Philosophy is valuable for pointing us in the right direction and helping us to think clearly. Sometimes philosophy can reveal a problem where nobody thought there was one, and this can lead to a new insight. Sometimes philosophy can identify and cure fallacies in reasoning. In solving a problem, it can highlight alternative solutions that might not have been noticed otherwise. But ultimately, physicists only tend to turn to philosophy when they have run out of ideas, and most of the time the connection of philosophy to practical matters seems tenuous at best. If philosophers have a weakness, it is only that they tend to think too much, whereas a physicist only thinks as hard as he needs to in order to get results.

After that brief detour, we are ready to return to our hero — physicist Percy Bridgman — and witness his own personal fling and falling-out with philosophy. In a previous post, we introduced Bridgman’s idea of operationalism. Recall that Bridgman emphasized that a physical quantity such as `length’ or `temperature’ should always be attached to some clear notion of how to measure that quantity in an experiment. It is not much of a leap from there to say that a concept is only meaningful if it comes equipped with instructions of how to measure it physically.

Although Bridgman was a physicist, his idea quickly caught on amongst philosophers, who saw in it the potential for a more general theory of meaning. But Bridgman quickly became disillusioned with the direction the philosophers were taking as it became increasingly clear that operationalism could not stand up to the demanding expectations set by the philosophers.

The main culprits were a group of philosophers called the Vienna Circle[2]. Following an idea of Ludwig Wittgenstein, these philosophers attempted to define concepts as meaningful only if they could somehow be verified in principle, an approach that became known as Verificationism. Verificationism was a major theme of the school of thought called `logical empiricism’ (aka logical positivism), the variants of which are embodied in the combined work of philosophers in the Vienna Circle, notably Reichenbach, Carnap and Schlick, as well as members outside the group, like the Berlin Society.

At that time, Bridgman’s operationalism was closely paralleled by the ideas of the Verificationists. This was unfortunate because around the middle of the 20th century it became increasingly apparent that there were big philosophical problems with this idea. On the physics side of things, the philosophers realized that there could be meaningful concepts that could not be directly verified. Einstein pointed out that we cannot measure the electric field inside a solid body, yet it is still meaningful to define the field at all points in space:

“We find that such an electrical continuum is always applicable only for the representation of electrical states of affairs in the interior of ponderable bodies. Here too we define the vector of electric field strength as the vector of the mechanical force exerted on the unit of positive electric quantity inside a body. But the force so defined is no longer directly accessible to experiments. It is one part of a theoretical construction that can be correct or false, i.e., consistent or not consistent with experience, only as a whole.”[1]

Incidentally, Einstein got this point of view from a philosopher, Duhem, who argued that isolated parts of a theory are do not stand as meaningful on their own, but only when taken together as a whole can they be matched with empirical data. It therefore does not always make sense to isolate some apparently metaphysical aspect of a theory and criticize it as not being verifiable. In a sense, the verifiability of an abstract quantity like the electric field hinges on its placement within a larger theoretical framework that extends to the devices used to measure the field.

In addition, the Verificationists began to fall apart over some rather technical philosophical points. It went something like this:

Wittgenstein: “A proposition is meaningful if and only if it is conceivablefor the proposition to be completelyverified!”

Others: “What about the statement `All dogs are brown’? I can’t very well check that all dogs are brown can I? Most of the dogs who ever lived are long dead, for a start.”

Wittgenstein: “Err…”

Others: “And what about this guy Karl Popper? He says nothing can ever be completely verified. Our theories are always wrong, they just get less wrong with time.”

Wittgenstein: *cough* *cough* I have to go now. (runs away).

Carnap: Look, we don’t have to take such a hard line. Statements like `All dogs are brown’ are still meaningful, even though they can’t be completely verified.

Schlick: No, no, you’ve got it wrong! Statements like `All dogs are brown’ are meaningless! They simply serve to guide us towards other statements that do have meaning.

Quine: No, you guys are missing a much worse problem with your definition: how do you determine which statements actually require verification (like `The cat sat on the mat’), and which ones are just true by definition (`All bachelors are unmarried’)? I can show that there is no consistent way to separate the two kinds of statement.

(Everybody’s head explodes)

So you can see how the philosophers tend to get carried away. And where was poor old Percy Bridgman during all this? He was backed into a corner, with people prodding his chest and shouting at him:

Gillies: “How do you tell if a measurement method is valid? If there is nothing more to a concept than its method of measurement, then every method of measurement is automatically valid!”

Bridgman: “Well, yes, I suppose…”

Positivists: “And isn’t it true that even if we all agree to use a single measurement of length, this does not come close to exhausting what we mean by the word length? How disappointing.”

Bridgman: “Now wait a minute –”

Margenau: “And just what the deuce do you mean by `operations’ anyhow?”

Bridgman: “Well, I … hey, aren’t you a physicist? You should be on my side!”

(Margenau discreetly melts into the crowd)

To cut a long story short, by the time Quine was stomping on the ashes of what once was logical empiricism, Bridgman’s operationalism had suffered a similar fate, leaving Bridgman battered and bloody on the sidelines wondering where he went wrong:

“To me now it seems incomprehensible that I should ever have thought it within my powers … to analyze so thoroughly the functioning of our thinking apparatus that I could confidently expect to exhaust the subject and eliminate the possibility of a bright new idea against which I would be defenseless.”

To console himself, Bridgman retreated to his laboratory where he at least knew what things were, and could spend hours hand-drilling holes in blocks of steel without having to waste his time arguing about it. Sometimes the positivists would prod him, saying:

“Bridgman! Hey Bridgman! If I measure the height of the Eiffel tower, does that count as an operation, or do you have to perform every experiment yourself?” to which Bridgman would narrow his eyes and mutter: “I don’t trust any experimental results except the ones I perform myself. Now leave me alone!”

Needless to say, Bridgman’s defiantly anti-social attitude to science did not help improve the standing of operationalism among philosophers or physicists; few people were prepared to agree that every experiment has to be verified by an individual for him or herself. Nevertheless, Bridgman remained a heroic figure and a defender of the scientific method as the best way to cope with an otherwise incomprehensible and overwhelming universe. Bridgman’s stubborn attitude of self-reliance was powerfully displayed in his final act: he committed suicide by gunshot wound after being diagnosed with metastatic cancer. In his suicide note, he wrote [3]:

“It isn’t decent for society to make a man do this thing himself. Probably this is the last day I will be able to do it myself.”

Bridgman’s original conception of operationalism continues to resonate with physicists to this very day. In the end he was forced to admit that it did not constitute a rigorous philosophical doctrine of meaning, and he retracted some of his initially over-optimistic statements. However, he never gave up the more pragmatic point of view that an operationalist attitude can be beneficial to the practicing scientist. Towards the end of his life, he maintained that:

“…[T]here is nothing absolute or final about an operational analysis […]. So far as any dogma is involved here at all, it is merely the conviction that it is better, because it takes us further, to analyze into doings or happenings rather than into objects or entities.”

“The main point is that sending the vacuum state is not nothing”. –Nicolas Gisin

I recently stumbled upon the following delightful exchange in the literature. It began with a tantalizing paper by Salih et. al., published in Physical Review Letters, in which the authors claim to have found a way to communicate information from one place to another without physical particles being transmitted in between. At first this seems astounding: surely information must be communicated by means of physical systems? Okay, some of you wiseguys will complain that you can transmit information by sending a wave through a medium without any individual of the particles making the entire journey, but that is not the point — one molecule must carry the influence to the next molecule, who carries it to the next, and so on. But Salih et. al. claimed to have found a way to transmit a message using photons and linear optics such that ultimately there is no complete chain of relays connecting the source to the detector. The authors write:

“In summary, we strongly challenge the longstanding assumption that information transfer requires physical particles to travel between sender and receiver by proposing a direct quantum communication protocol whereby, in the ideal asymptotic limit, no photons pass through the transmission channel, thus achieving complete counterfactuality. In so doing we highlight the essential difference between classical and quantum information.”

At this point, you are in danger of falling into a trap. Assuming that you believe the result of the paper (you should — it is correct) then you might chalk it up to yet another example of `quantum weirdness’. After all, the protocol is based on the `counterfactual’ nature of quantum mechanics, whereby one can gather information about a system without collapsing its wavefunction. This is precisely the phenomenon at work in the fascinating quantum Zeno effect (merely watching an atom stops it from decaying) and the ultimate in quantum bizarreness, the Elitzur-Vaidman Bomb Test, where it is possible to detect the presence of a bomb using a photon that doesn’t go anywhere near the bomb (the point is that it might have done). So is it any wonder that we can now tack another feat to the list, namely the phenomenon of `quantum counter-factual communication’?

We must be careful, however! If it walks like a quantum and talks like a quantum, it might still turn out to be qwassical. Indeed, in this paper published in Rapid Communications, Nicolas Gisin showed that you can communicate by `sending nothing’, without needing quantum mechanics at all! Actually, you can do it quite easily using just office stationary and the person sitting next to you. Here’s how:

“Hey Bob, at 12:00 I’m going to check the results of the live football match. If my team won, I’ll let you know by throwing a pencil at your head. If my team lost, I won’t throw anything.”

Sure enough, at the appointed hour, Bob ducks reflexively, but to no purpose: there is no impending pencil. Nevertheless, he now knows that your team lost the game. Okay, perhaps this is cheating a little bit. After all, in the event that your team won, a physical object really would have to traverse the length of the office and strike Bob on the head. So in some sense, the information still depends on the transmission of the object.

But wait! Christine is sitting midway between yourself and Bob and she offers to help out. Instead of communicating directly with Bob, you establish the routine with Christine. If she receives nothing from you at 12:00, she throws her stapler at Bob. If she receives your pencil, she does nothing. Now, if your team loses, a stapler goes from Christine to Bob but nothing from you to Christine. And if your team wins, a pencil goes to Christine but nothing to Bob. Either way, no physical object traverses the distance from you to Bob, and yet by 12:01 he knows which team won. Astounding!

What has happened here? Have we demonstrated that Rolf Landauer was wrong — that information is not tied to physical systems after all? Hardly. We have merely neglected the one particular exchange of physical systems that makes this whole thing possible: the channel itself. In his tactfully amusing one-and-a-half page note, Gisin drives this point home using blue and red balls instead of pencils and staplers, but the gist is the same.

For those of you who are still mystified, I have composed the following fairy-tale to bring the physics directly to your inner child.

Once upon a time, in a Kingdom far, far away, there was a King with a very beautiful daughter. Unfortunately the King’s castle was located some distance from the nearest town, where all the most eligible bachelors resided. The only means of communication was the mail-delivery cart and the King’s personal messenger — a small carrier pigeon with a black mark under its eye, identifying it as the Royal emissary. While the mail cart dealt with regular mail, the King reserved his pigeon for gathering news about the stock market.

Day after day, the postman came from the village, his cart practically weighed down by petitions and gifts from the local men for the hand of the Princess. Eventually the King got sick of it, so he made an announcement: no more letters by mail. Each suitor was to apply sequentially, in an order determined by drawing names out of a hat, by writing a letter to the King deliverable by the Royal carrier pigeon. And the King said: “Whoever is able to write me a letter, brought to me by the Royal pigeon, such that reading the letter does not reveal to me its informational contents, shall have my daughter’s hand. Should you fail in the task, you will be beheaded. That is all.”

Naturally, the less brilliant and more cowardly portions of the would-be suitors gave up immediately. But a few brave souls placed their names in the hat. The first bloke was a lawyer and immediately sent off a piece of paper to the King that was completely blank, and spent a good amount of time bragging about his cleverness to his friends afterwards over beer. The King turned up in town the next day.

“Would you agree,” said the King, “that nobody has informed me what you wrote in the letter?”

“Quite,” said the lawyer, who knew the King well enough to know he would not so blatantly cheat on such matters.

“Well then. If I were to say that you wrote nothing at all in the letter, you would have to agree that I was able to deduce this fact by reading your letter. Not so?”

Seeing where things were headed, the lawyer immediately tried to flee, but they caught him and lopped off his head as promised. Nobody could deny that the informational content of the letter – namely, that it was blank – could be deduced immediately by reading it.

At this point, the remaining eligible bachelors swamped the town hall and demanded to have their names removed from the hat. Only one name remained: that of the King’s stockbroker, who was unhappily out of town that day. By the time he returned and realized the fate that undoubtedly awaited him, it was far too late to request his name be removed. He had no choice but to go through with it.

Luckily for him, it happened that the Princess had seen him at business lunches with her father on more than a few occasions, and found him both witty and attractive, despite his nervous demeanor, which was a hallmark of all stockbrokers in those troubled times. Being rather a bright spark, she sent him a secret letter by carrier-squirrel outlining a clever plan. When he received her letter, the stockbroker immediately grabbed a bottle of ink and ran to the baker to fetch a sack of breadcrumbs.

Finally, the appointed day arrived. The King had a replacement broker standing by, the executioner’s axe was sharpened and a hush was upon the town. The King stepped out onto his balcony in the fresh light of morning to receive the letter. Sure enough, with the familiar fluttering of wings the pigeon arrived with a parchment bearing the broker’s seal. In the letter was the usual information about stocks – the king noted some strange selling happening in the housing sector – and a polite how-do-you-do. How boring! He thought. But then he noticed two other pigeons that had been quietly waiting on the perch, which he hadn’t seen in his haste. Each of the also bore the royal mark below their right eye. Tearing open the letters, the King found similar contents, but with the stock information slightly different here or there, and the greeting rephrased one way or another. By this time more and more pigeons were arriving, all identical, all bearing letters from the stockbroker, but each with different contents.

“Damnation!” cried the King, amidst a growing swarm of pigeons. “Which one is the real Royal pigeon? I want to know what the stock market really did today!” Of course, one of the pigeons was the real pigeon and the letter it carried did contain the information the King desired. In fact he had even read it — it was the first one. But in the end he still had to go into town to ask the stockbroker what he had written. And that’s how the humble stockbroker avoided the stocks and remained on the market for the fair Princess.

The moral of the story is, if you have established a communication channel, you can’t help but send information along it. There is always one value that you designate as `vacuum’, which is conventionally the state that gets sent when the source is not being manipulated in any particular way, such as the blank piece of paper in the story, or the empty air that constantly blows from the window to your desk to Bob’s desk at all hours of the day, gently bombarding his head. But the blank page still needs to be sent, in precisely the manner arranged, in order for it to carry information. Only by destroying the channel itself (such as by drowning it in noise) can you prevent the transmission of information. `Vacuum’ means no action, but `nothing’ means no channel at all; the vacuum is not nothing.